Q89 / 160 · AP EAMCET 2020 (17 Sep Shift 2) · Mathematics

For \(f(x)=\frac{\sin \pi[x]}{1+[x]}+\frac{x}{2+3 x}\), where \([x]\) denotes the greatest integer function, the domain and range in \(R\) are respectively

  • A
    \(R-\left\{-1, \frac{-2}{3}\right\}\) and \(R-\left\{\frac{1}{3}\right\}\)
  • B
    \(\mathrm{R}-\left\{-1, \frac{-2}{3}\right\}\) and \([-1,1]\)
  • C
    \(R-[-1,0)\) and \(R-\left\{\frac{1}{3}\right\}\)
  • Correct answer
    D
    \(\mathrm{R}-[-1,0)\) and \([-1,1]\)

Solution

As \([x]=-1\) when \(x \in[-1,0)\). This makes denominator of first part \(1+[x]=0\). Hence, Interval \([-1,0)\) must be excluded from domain set. \(\therefore \mathrm{D}(f)=R-[-1,0)\) Also at \(x=0\) (which is part of domain), value of function is zero. i.e. \(f(0)=0\) So option (d) is correct.

Hence, the correct answer is option: 'D'.

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